Project Gutenberg's Catalan's Constant [Ramanujan's Formula], by Greg Fee

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Title: Catalan's Constant [Ramanujan's Formula]

Author: Greg Fee

Posting Date: August 13, 2008 [EBook #682]
Release Date: October, 1996

Language: English

Character set encoding: ASCII

*** START OF THIS PROJECT GUTENBERG EBOOK CATALAN'S CONSTANT ***

Produced by Greg Fee

Catalan's Constant [Ramanujan's Formula]

Catalan constant to 300000 digits computed on September 29, 1996
by using a Sun Ultra-Sparc in 1 day 8 hour 15 min 15 sec 55 hsec.
The algorithm used is the standard series for Catalan, accelerated
by an Euler transform. The algorithm was implemented using the LiDIA
library for computational number theory and it is part of the
multiprecision floating-point arithmetic of the package.

LiDIA is available from

ftp://crypt1.cs.uni-sb.de/pub/systems/LiDIA/LiDIA-1.2.1.tgz
http://www-jb.cs.uni-sb.de/LiDIA/linkhtml/lidia/lidia.html

The implementation of the algorithm is:

inline void
const_catalan (bigfloat & y)
{
bigfloat p;
bigfloat t;
int i = 1, j = 3; // j = 2*i+1

// y = t = p = 1/2
divide (y, 1, 2);
t.assign (y);
p.assign (y);

// while t is greater than the desired accuracy
while (!t.is_approx_zero ())
{
// do
// p = p * (i/j);
// t = (t * i + p) / j;
// y = y + t;
// i++; j+=2;
multiply (p, p, i);
divide (p, p, j);
multiply (t, t, i);
add (t, t, p);
divide (t, t, j);
add (y, y, t);
i++;
j += 2;
}
}

Here is the output of the program:

Calculating Catalan's constant to 300000 decimals

Time required: 1 day 8 hour 15 min 15 sec 55 hsec

--------------------------------------------------------------------------

Additional REFERENCES:

Catalan constant is: sum((-1)**(n+1)/(2*n-1)**2,n=1..infinity) also known
under the name beta(2), see ?catalan in Maple for more details.

The previous record was 200000 digits, also from Thomas Papanikolaou
and before that: 100000 digits
was due to Greg Fee and Simon Plouffe on August 14, 1996,